Group Summary: Quasi-Equilibrium Turbulent Flows

Motivated by the importance of stratified shear flows in geophysical and environmental circumstances, we characterize their energetics, mixing and spectral behaviour through a series of direct numerical simulations of turbulence generated by Holmboe wave instability (HWI) under various initial conditions. We focus on circumstances where the stratification is sufficiently ‘strong’ so that HWI is the dominant primary instability of the flow. Our numerical findings demonstrate the emergence of self-organized criticality (SOC) that is manifest as an adjustment of an appropriately defined gradient Richardson number, $Ri_{g}$, associated with the horizontally averaged mean flow, in such a way that it is continuously attracted towards a critical value of $Ri_{g}\sim 1/4$. This self-organization occurs through a continuously reinforced localization of the ‘scouring’ motions (i.e. ‘avalanches’) that are characteristic of the turbulence induced by the breakdown of Holmboe wave instabilities and are developed on the upper and lower flanks of the sharply localized density interface, embedded within a much more diffuse shear layer. These localized ‘avalanches’ are also found to exhibit the expected scale-invariant characteristics. From an energetics perspective, the emergence of SOC is expressed in the form of a long-lived turbulent flow that remains in a ‘quasi-equilibrium’ state for an extended period of time. Most importantly, the irreversible mixing that results from such self-organized behaviour appears to be characterized generically by a universal cumulative turbulent flux coefficient of $\unicode[STIX]{x1D6E4}_{c}\sim 0.2$ only for turbulent flows engendered by Holmboe wave instability. The existence of this self-organized critical state corroborates the original physical arguments associated with self-regulation of stratified turbulent flows as involving a ‘kind of equilibrium’ as described by Turner (1973, Buoyancy Effects in Fluids, Cambridge University Press).

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Mean and turbulence dynamics in unsteady Ekman boundary layers

Journal of Fluid Mechanics ◽

10.1017/jfm.2017.76 ◽

2017 ◽

Vol 816 ◽

pp. 209-242 ◽

Cited By ~ 15

Author(s):

Mostafa Momen ◽

Elie Bou-Zeid

Keyword(s):

Turbulent Flows ◽

Time Scale ◽

Quasi Equilibrium ◽

Mean Flow ◽

Turbulent Friction ◽

Friction Forces ◽

Unsteady Pressure ◽

Closure Models ◽

Complex Interactions ◽

The Mean

Unsteady pressure gradients in turbulent flows not only influence the mean, but also affect the higher statistical moments of turbulence. In these flows, it is important to understand if and when turbulence is in quasi-equilibrium with the mean in order to better capture the dynamics and develop effective closure models. Therefore, this study aims to elucidate how turbulence decays or develops relative to a time-varying mean flow, and how the turbulent kinetic energy (TKE) production, transport and dissipation respond to changes in the imposed pressure forcing. The focus is on the neutral unsteady Ekman boundary layer, where pressure-gradient, Coriolis and turbulent friction forces interact, and the analyses are based on a suite of large-eddy simulations with unsteady pressure forcing. The results indicate that the dynamics is primarily controlled by the relative magnitudes of three time scales: the inertial time scale (characterized by Coriolis frequency${\sim}12$ hours at mid-latitudes), the turbulent time scale (${\sim}2$ hours for the largest eddies in the present simulations) and the forcing variability time scale (which is varied to reflect different (sub)meso to synoptic scale dynamics). When the forcing time scale is comparable to the turbulence time scale, the quasi-equilibrium condition becomes invalid due to highly complex interactions between the mean and turbulence, the velocity profiles manifestly depart from the log-law and the normalized TKE budget terms vary strongly in time. However, for longer, and surprisingly for shorter, forcing times, quasi-equilibrium is reasonably maintained. The analyses elucidate the physical mechanisms that trigger these dynamics, and investigate the implications on turbulence closure models.

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